Time Quasi-periodic unbounded perturbations of Schr\"odinger operators and KAM methods

Abstract

We eliminate by KAM methods the time dependence in a class of linear differential equations in $\ell^2$ subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of the Floquet spectrum of the operator $ H_0+\epsilon P(\om t)$ for $\epsilon$ small. Here $H_0$ is the one-dimensional Schr\"odinger operator $p^2+V$, $V(x)\sim |x|^{\alpha}, \alpha >2$ for $|x|\to\infty$, the time quasi--periodic perturbation $P$ may grow as $\displaystyle |x|^{\beta}, \beta <(\alpha-2)/{2}$, and the frequency vector $\omega$ is non resonant. The proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on Kuksin's estimate of solutions of homological equations with non constant coefficients.